Some more recently discovered (for me) YouTube closed caption options.
If you click the three dot menu under the currently playing video...
...and that video has closed captions, you should see a 'Show transcript' option in that menu, click that and all the closed captions should show in a transcript window beside or below the video.
The captions will highlight in sequence as the video plays and if you click any one of the captions, the video will jump to that position. There is a three dot menu at the top right of the transcript window where you can toggle the timestamps on and off, I guess that makes it better for copy pasting the captions should you want to.